Dynamic light scattering

In section 4.2 we have seen that with inelastic light scattering, where we measure with spec-troscopic methods the frequency power spectrum of the auto correlation function, we can observe dynamics from optical frequencies (10¹⁴ Hz) down to some 10⁶ - 10⁷ Hz. With neutron spin echo spectroscopy, this frequency range can be extended by about one order of magnitude. In this section we want to ask the question how we can observe even slower dynamics.

The answer is to measure the time evolution of the instantaneous correlation function, i. e. by measuring an integral spectrum at a given time all fast processes have relaxed and we are left with the slow dynamics. In dynamic light scattering, one measures the normalised intensity auto correlation function (IACF):

( ) ( ) ( )

^{Q,t : IQ,0 IQ,t /I2}

gˆI = ⋅ . (51)

In practise the detector is positioned at a given scattering angle corresponding to a momentum transfer Q. The intensity fluctuations in the detector are determined and an auto correlation function is calculated according to appendix A1. This auto correlation function is normalised to the square of the average intensity I in the detector. However, in section 3.3 and 4.1, we have learnt that the quantity of interest is actually the auto correlation function of the electric field (EACF). The normalised EACF is given by

( )

^E_S

( ) ( )

^{Q,0 E}_S^{Q,t /I}

µF0 2 : 1 t , EQ

gˆ ε ⋅ ∗

= . (52)

Since the electric field ^E_S^∗

( )

^Q,t is related to the scattering power density via the Fourier transform, the EACF is related to the intermediate scattering function. It can be shown that the intensity auto correlation function and the electric field auto correlation function are con-nected by the so-called Siegert relation:

( )

^{Q,t 1 gˆ}_E

( )

^Q,t

gˆI = + . (53)

Thus by an experimental determination of the intensity auto correlation function, we get ac-cess to the Fourier transform of the density auto correlation function, which is the quantity of interest.

Let us discuss an example for photon correlation spectroscopy. We have chosen the scattering of coherent 8.2 keV x-rays from a synchrotron radiation source by porous silica gel. When the sample is illuminated by the coherent x-ray beam, we observe on the area detector a so-called speckle pattern (figure 22), which is due to the instantaneous density modulations within this sample.

Fig. 22: Speckle pattern from an aerogel recorded with a CCD detector using coherent x-rays [8]. The speckle structure becomes better visible, if one pixel row in the horizontal direction is plotted (continuums line in the bottom picture).

For an inhomogeneous sample with a density distribution constant in time, we will observe a static speckle pattern. However for a sample, which has internal slow dynamics, the speckle

pattern will change with time and we can observe the intensity in a given direction with a small pinhole or within one pixel of an area detector. From the Siegert relation, we can deduce the time dependence of the intermediate scattering function. This function is depicted for colloidal silica, suspended in a solvent (water glycerol mixture) in figure 23.

S(Q,t)

Fig. 23: Intermediate scattering function measured by x-ray photon correlation spectroscopy XPS (open circles) and optical photon correlation spectroscopy (closed squares) for the same momentum transfer Q = 9⋅10^-4 Å^-1 in an optically opaque sample of colloidal silica [8].

An exponential decay can be fitted to the XPCS measurement and the translational diffusion coefficient can be determined. Figure 22 shows that the PCS intermediate scattering function relaxes faster as compared to the XPCS data. The authors attribute this effect to multiple scattering events, which are present in the optical experiment.

5 Summary

An introduction into the investigation of soft condensed matter systems by scattering methods has been given. We have seen that neutrons, light and x-rays are particularly useful for such studies. However independent of the probe, coherent scattering always measures pair correla-tion funccorrela-tions and the scattering cross seccorrela-tion is proporcorrela-tional to the spatial and temporal Fou-rier transform of these correlation functions.

Condensed matter investigations face the problem that relevant lengths and time scales cover many orders of magnitude. Figure 24 gives an overview over the characteristic time scales found in condensed matter investigations. These time scales range from the fs regime for fast electronic excitations up to macroscopic time scales for relaxation processes in glasses or spin glasses.

Fig. 24: Characteristic time scales found in condensed matter research.

10-15 fs

10-12 ps

10-9 ns

10-6 µs

10-3 ms

100 s

103 ks

L_III E

V(φ)

1 s 1000 slog t log M

photo effect plasmons

magnons, phonons diffusion

rotational tunnelling

spin glass relaxation characteristic

time scales

γ

polymer-reptation

With appropriate scattering methods, all these processes on different time scales can be in-vestigated. Which scattering method is appropriate for which region within the "scattering vector Q - energy E plane" is plotted schematically in figure 25. A scattering vector Q corresponds to a certain length scale, an energy to a certain frequency, so that the characteristic lengths and times scales for the various methods can be directly determined from figure 25.

10

^-2

10

⁰

10

²

10

⁴

10

⁶

10

⁸

10

¹⁰

10

¹²

10

¹⁴

10

¹⁶

10

^-16

10

^-14

10

^-12

10

^-10

10

^-8

10

^-6

10

^-4

10

^-2

10

⁰

10

^-6

10

^-5

10

^-4

10

^-3

10

^-2

10

^-1

10

⁰

10

¹

10

²

10

³

10

⁴

10

⁵

10

⁶

frequency ν [Hz] energy E=h ν [eV]

length d=2π/Q [Å]

Optical photon- correlation-spectroscopy

PCS time resolved x-ray and neutron scattering optical

Raman-spectroscopy optical

Brillouin-spectroscopy

scattering vector Q [Å

^-1

]

neutron and x-ray scattering

x-ray photon correlation spectroscopy

XPCS

Fig. 25: Regions in frequency v and scattering vector Q or energy E and length d plane, which can be covered by various scattering methods.

Appendices

A Autocorrelation Functions

Let us assume a quantity A that fluctuates in time as depicted in the insert of figure A1.

time → τ

Fig. A1: The insert gives an example of a quantity A(t) that fluctuates in time. The main figure shows the general behaviour of the autocorrelation function. It has a maxi-mum of <A²> for τ = 0. From this maximum, it decays with a time constant τA to-wards the asymptotic value of <A>² for infinite times.

For this quantity, we can define the average value

∫

( )

func-tion A(t) varies monotonically, A(t+τ) approaches the values of A(t) for small values of τ, which means that both are correlated in time. Similar correlations can occur at larger times, e. g. for quasi-periodic fluctuations. As a measure of this correlation, the autocorrelation function is introduced:

A 2

This autocorrelation function has a very general property depicted in figure A1: it decays from a maximum value of <A²> for τ = 0 to the asymptotic value of <A>² for infinite times with a characteristic time constant of τA. Often times, this decay follows a simple exponential law exp(−τ/τ_A).

References

1. "Neutron Scattering",

Schriften des Forschungszentrums Jülich: Materie und Material; Vol. 9 2. "Femtosekunden und Nano-eV: Dynamik in kondensierter Materie",

Schriften des Forschungszentrums Jülich: Materie und Material; Band 3 3. John M. Cowley, "Diffraction Physics", North Holland Amsterdam, 1990

4. Steven W. Lovesey, "Theory of neutron scattering from condensed matter", Clarendon Press, Oxford, 1987

5. J. K. G. Dhont, "Light Scattering", chapter 3 in: "An Introduction to Dynamics of Colloids", Elsevier, Amsterdam (1996)

6. A. Messiah, "Quantenmechanik", Band 1 + 2, Walter de Gruyter, Berlin (1976) 7. J. L. Yarnell, M. J. Cats, R. G. Wenzel, S. H. Koenig, Phys. Rev. A 7 (1973), 2130 8. D. O. Riese et al., Phys. Rev. Lett. 85 (2000), 5460

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